High-Power Analogues of the Turán-Kubilius Inequality, and an Application to Number Theory

1980 ◽  
Vol 32 (4) ◽  
pp. 893-907 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Number Theory ◽  
Absolute Constant ◽  
Arithmetic Function ◽  
Standard Form ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Coprime Integers ◽  
Complete Catalogue ◽  
Image Position ◽  
Complex Valued

An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we defineA standard form of the Turán-Kubilius inequality states that(1)holds for some absolute constant c1, uniformly for all complex-valued additive arithmetic functions ƒ (n), and real x ≧ 2. An inequality of this type was first established by Turán [11], [12] subject to some side conditions upon the size of │ƒ(pm)│. For the general inequality we refer to [10].This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [3], [4], and [1].

The law of large numbers for additive arithmetic functions

1975 ◽  
Vol 78 (1) ◽  
pp. 33-71 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Law Of Large Numbers ◽  
Arithmetic Function ◽  
Sufficient Conditions ◽  
Growth Conditions ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Large Numbers ◽  
Coprime Integers ◽  
Weak Growth ◽  
Image Position

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequenciesWe shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

General asymptotic distributions for additive arithmetic functions

1976 ◽  
Vol 79 (1) ◽  
pp. 43-54 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Arithmetic Function ◽  
Asymptotic Distributions ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Additive Arithmetic Function ◽  
Image Position

1. Let f(n) be a real-valued additive arithmetic function. Let α(x) and β(x) > 0 be real valued functions, defined for x ≥ 2. Define the frequencies

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

On average values of arithmetic functions

1941 ◽  
Vol 37 (4) ◽  
pp. 358-372 ◽  
Author(s):  
E. Fogels
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Average Order ◽  
Image Position

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we haveandas x → ∞.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

An Extremal Problem in Number Theory

1967 ◽  
Vol 10 (2) ◽  
pp. 173-177 ◽  
Author(s):  
H. L. Abbott ◽  
B. Gardner
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Extremal Problem ◽  
Absolute Constant ◽  
Greatest Common Divisor ◽  
Image Position

Let n and k be integers with n ≥ k ≥ 3. Denote by f(n, k) the largest positive integer for which there exists a set S of f (n, k) integers satisfying (i) and (ii) no k members of S have pairwise the same greatest common divisor. The problem of determining f(n, k) appears to be difficult. Erdős [2[ proved that there is an absolute constant c > 1 such that for every ∈ > 0 and every fixed k1provided n > no (k, ∈). In [l[ it i s proved that for every ∈ > 0 and every fixed k

Additive arithmetic functions on intervals

1988 ◽  
Vol 103 (1) ◽  
pp. 163-179 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Arithmetic Function ◽  
Value Distribution ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Positive Integers

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.

Identities for Multiplicative Functions

1967 ◽  
Vol 10 (1) ◽  
pp. 65-73 ◽  
Author(s):  
M. V. Subbarao ◽  
A. A. Gioia
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Prime Divisors ◽  
Image Position

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also letWe recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.

scholarly journals A study on some generalized multiplicative and generalized additive arithmetic functions

2021 ◽  
Vol 27 (1) ◽  
pp. 32-44
Author(s):  
D. Bhattacharjee ◽  
Keyword(s):  
Additive Function ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Positive Integers

In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.

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